If I trained a model to perform semantic segmentation on images with only one object, would it also work on images with multiple objects?
Without experimental evidence to back me up, I can not answer this with 100% confidence. However, I am fairly certain that this will cause issues depending on the model.

U-net is essentially an auto-encoder, and due to the fact that it is all just one big neural network, it is likely it will learn the easiest pattern (as all NN do), and that is to find one single instance of an object and shade that region.

Now why does this depend on the model? Well let's say you are using something slightly different, where region proposals are generated by a determinisitc algorithm we've predefined, then these regions are run through a CNN to segment them. In this case, as each region is without context of the entire image, the difference between 2 objects in an image and 1 is indistinguishable to the network (as regions may overlap), and as such, only using images with 1 object will not pose any problems (there is a name for early models like these, though it escapes me).

So assuming I am correct, what should you do? The models that use a deterministic algorithm for region proposals are slow and old, so I wouldn't suggest that. Instead, I would think that you should first do some testing, to see if it actually does cause issues. Assuming it does, a good option could be to tamper with the training data and separate segments by a few pixels to sort of "force" multiple objects into existence.

Regardless of such, I would still suggest using U-net. Fixing this issue (if it does arrise) should be relatively easy to do, so there's little to lose by using U-net and just trying the training.